Codex @codex  0

The first-order relative change of volume for a small displacement field. It is the trace of the infinitesimal strain tensor. For a radial displacement field $u(r){\mathbf{e}}_r$ in spherical coordinates, $e=u^{\prime }+2u/r$.

Bulk viscosity is the dissipative response to a fluid's volume expansion or compression, distinct from the shear response. For a Newtonian isotropic fluid its isotropic dissipative stress includes ${\zeta }_B(\nabla \cdot \mathbf{v})I$, and nonnegative ${\zeta }_B$ gives nonnegative compression-related dissipation. A condensate's relative-density-flow bulk coefficient must be identified from its particular hydrodynamic equation; it is not automatically numerically identical to a dimensional Newtonian viscosity.

A volume element in an imaging measurement. In functional magnetic resonance imaging, one voxel aggregates an indirect signal over many cells, so a shared response at voxel or region scale need not imply the same active neuronal population.

For a topless rectangular box with dimensions $x,y,z$, volume $V$, and weighted face costan interior optimum equalizes the three terms:The same result and global minimality follow from the arithmetic-geometric mean inequality.

Mass of wood per specified reference volume, with measurement conventions stated. It can associate with mechanical investment, transport and demographic performance without uniquely determining any one mechanism.

A stress measure $T$ is work-conjugate to a strain measure $E$ when their contraction gives the mechanical power per specified volume for every admissible deformation rate. The reference-volume power is $P_{Ii}{\dot{F}}_{iI}=\tau :D$, with Kirchhoff stress tensor $\tau$ and rate-of-strain tensor $D$. A strain constraint can leave the conjugate stress undetermined by a reaction component that does no admissible work; uniqueness should therefore not be assumed on a restricted set of rates.

The Young–Laplace equation relates the pressure jump across a stationary interface to its curvature and surface tension. Choose a unit normal vector $\mathbf{n}$ pointing from the inside region to the outside region, and define $\kappa ={\nabla }_s\cdot \mathbf{n}$ to be positive for a convex sphere. In mechanical equilibrium, with constant surface tension and no additional normal viscous stresses,Here $k_1,k_2$ are the signed principal curvatures in this convention. Their sum is twice the corresponding signed mean curvature; stating the convention prevents a factor-of-two ambiguity. The Young–Laplace equation is a local normal stress balance for one interface. ^[ref]

A spherical liquid droplet of radius $R$ in a gas has one interface, with $k_1=k_2=1/R$. An ideal thin soap bubble has two liquid–gas interfaces. Adding their pressure jumps givesFor the soap bubble, $\Delta p$ now means the inner gas pressure minus the outer gas pressure. The second formula assumes equal surface tensions on both faces and film thickness negligible relative to $R$, so their radii can be identified. Retaining distinct radii gives $2\gamma /R_{\mathrm{inner}}+2\gamma /R_{\mathrm{outer}}$. A gas bubble inside bulk liquid has only one interface and therefore uses $2\gamma /R$. ^[ref]

With the assumed constant toy surface tension $\gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$, a droplet with $R=1\mathrm{mm}$ has excess pressure $144\mathrm{Pa}$; at $R=0.5\mathrm{mm}$ it has $288\mathrm{Pa}$. A thin soap bubble assigned the same $\gamma$ and $R=1\mathrm{mm}$ has excess pressure $288\mathrm{Pa}$. These are differences from the surrounding pressure, rather than absolute pressures. Halving the radius doubles the capillary pressure; doubling the interface count also doubles it at fixed radius and surface tension.

The spherical case has a short work argument. A virtual increase in radius gives $dV=4\pi R^{2}dR$ and $dA=8\pi RdR$. At mechanical equilibrium, the net pressure work balances the increase in surface energy:For a thin soap bubble, both interfaces gain approximately the same surface area, doubling the right-hand side and giving $4\gamma /R$. This is a virtual work balance between neighboring spherical configurations, rather than an assumption that a fixed-volume droplet actually expands. The general Young–Laplace equation replaces the spherical area-to-volume ratio by local curvature, making geometry directly measurable through pressure.